Gibbs’ phenomenon and surface area

de Michele, L.; Roux, D.

doi:10.1090/S0002-9939-06-08639-4

Gibbs’ phenomenon and surface area
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by L. de Michele and D. Roux PDF

Proc. Amer. Math. Soc. 134 (2006), 3561-3566 Request permission

Abstract:

If a function $f$ is of bounded variation on $T^N\ (N\geq 1)$ and $\{{\varphi }_n\}$ is a positive approximate identity, we prove that the area of the graph of $f*{\varphi }_n$ converges from below to the relaxed area of the graph of $f$. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.

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